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%I #12 Nov 18 2024 22:31:19
%S 1,1,1,1,1,1,2,2,3,3,5,5,7,9,13,18,25,33,43
%N Number of real integers in n-th generation of tree T(r) defined in Comments, where r^2 = -r - 1 (i.e., r = (-1 + sqrt(3))/2).
%C Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.
%C See A274142 for a guide to related sequences.
%e If r = (-1 + sqrt(3))/2, then g(3) = {3,2r,r+1,r^2}, in which the number of real integers is a(3) = 1.
%t z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t u = Table[t[[k]] /. x -> (-1 + 3 I)/2, {k, 1, z}]; Table[
%t Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y Cf. A274142.
%K nonn,more,changed
%O 0,7
%A _Clark Kimberling_, Jun 13 2016